On the probabilistic modeling of consistency for iterated positional election procedures

A well-known fact about positional election procedures is that its ranking of m alternatives can change when some of the alternatives are removed from consideration—given a positional procedure on each of 2, 3, …, m alternatives and a collective preference order for each distinct subset of the m alternatives. Saari has established that with few exceptions, we can find a voter profile for which the collective preference order for each subset under the according positional procedure is the one given. However, Saari's results do not quantify the likelihood of finding such voter profiles. For small numbers of alternatives, William Gehrlein developed a statistical model to explore the probabilities that particular collective preference orders on subsets of alternatives can occur for large electorates. One goal of this research is to determine whether changes in the collective preference order as alternatives are removed can be considered to be the norm or an outlier for positional procedures.

This dissertation extends the research headed by Gehrlein in two directions. One, I generalize his statistical model to explore probabilities for iterated election procedures. Gehrlein's model previously produced results only for three alternatives and in limited cases for four alternatives. I have extended this model to produce results for up to five alternatives, including analysis of instant-runoff voting and runoff elections. Two, Gehrlein's model required specific conditions on the probability distribution of individual voter preferences across the population. I relax this assumption so that for any probability distribution of individual voter preferences across the population, I can explore the probability that a collective preference order is inconsistent with the outcomes when alternatives are removed. These results provide a foundation for discussing the impact of removing alternatives on elections across all large electorates. I also apply these results to two recent United States elections wherein a third-party candidate received a significant share of the votes: the 1992 U.S. Presidential election and the 1998 Minnesota Gubernatorial election. Overall, my research will suggest that as the number of alternatives increases, the likelihood of finding changes in the collective preference order as alternatives are removed will approach one.